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| Mirrors > Home > MPE Home > Th. List > itgeq2dv | Structured version Visualization version GIF version | ||
| Description: Equality theorem for an integral. (Contributed by Mario Carneiro, 7-Jul-2014.) |
| Ref | Expression |
|---|---|
| itgeq2dv.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) |
| Ref | Expression |
|---|---|
| itgeq2dv | ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | itgeq2dv.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) | |
| 2 | 1 | ralrimiva 3146 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) |
| 3 | itgeq2 25813 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥) | |
| 4 | 2, 3 | syl 17 | 1 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥) |
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